Converting energy into mass happens when you smash protons together so hard that it knocks the quarks out. The quark flying off stores energy in the bond being stretched like a rubber band, until it has stored enough energy that it can create more quarks.
Photons with high enough energy can create matter particles.
Bonding energy and potential energy within atomic and subatomic structures show up directly as measurable mass of the resulting system.
What "form"? If you create brand new particles, you get a balance of particles and anti-particles (though some particles are their own anti). Which particle you create is limited by the mass of the particle: you need enough energy to create it. And, it has to be a kind of particle that relates with the form of energy being applied. Within those rules, it's totally random.
If you add energy to an existing particle or system of particles, it just gets heavier.